Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #44446 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#553 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,19)(12,24)(13,28,29,17)(14,18)(15,21,32,22)(16,26,34,25)(20,36)(23,31)(27,33)(30,38)(35,37,40,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,21)(14,23)(16,30)(17,19)(18,32)(22,29)(24,35)(25,37)(26,39)(27,38)(28,31)(33,40)(34,36);(9,12,14,27)(10,16,18,34)(11,20,23,38)(13,26,29,25)(15,30,32,36)(17,24,28,33)(19,35,31,40)(21,39,22,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37687);